Rotate the result of a mirror modifier
Ben, no the mirror modifier only produces symmetry across a plane; there is no way to rotate. One approach I used to model a tire was to model a strip across the entire width of the tire, and use an array modifier around an empty to arrange the instances of the array around the center point. The bit of the tread I modeled is outlined in white.

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I should note that I only modeled the raised parts of the tread, and not the side wall or base; these were a separate object.

This technique worked, but the results were unsatisfactory because I was modeling a real tire, and it turned out that the different strips around the tread did not have the same size; the left three parts of the thread occupied a smaller part of the radius (1/48th of the radius, as I recall) than the right two parts (1/40th of the radius), so what I wound up doing is modeling the tread pattern as two different bits, the three left in on bit, the two right in the other, and applying the two different segments around the body (the sidewalls, and base of the tread), as in the illustration, where I've rotated the tire body, and highlighted the two segment arrayed object for illustrative purposes.